Automorphism groups of superextensions of groups

Abstract

The superextension λ(X) consists of all maximal linked families on X. Any associative binary operation * : X × X → X can be extended to an associative binary operation * colonλ(X) × λ(X) → λ(X). In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of order ≤ 5.

On the automorphism group of the superextension of a semigroup

“…The monogenic semigroup M 1,3 of index 1 is isomorphic to the cyclic group C 3 . The automorphism groups of superextensions of groups and finite monogenic semigroups and their ideals were described in [8] and [9]:…”

Automorphisms of semigroups of k-linked upfamilies

“…In this paper, we investigate the automorphism groups of the extensions N k (S) of a semigroup S. The thorough study of various extensions of semigroups was started in [13] and continued in [1][2][3][4][5][6][7][8][9][10][14][15][16][17][18][19]. The largest among these extensions is the semigroup υ(S) of all upfamilies on S. A family A of non-empty subsets of a set X is called an upfamily if for each set A ∈ A any subset B ⊃ A of X belongs to A.…”